Problem: $ 1.\overline{94} \div 0.\overline{2} = {?} $
Answer: First convert the repeating decimals to fractions. $\begin{align*} 100x &= 194.9494...\\ x &= 1.9494...\end{align*} $ $\begin{align*} 99x &= 193 \\ x &= \dfrac{193}{99}\end{align*} $ $\begin{align*} 10y &= 2.2222...\\ y &= 0.2222...\end{align*} $ $\begin{align*} 9y &= 2 \\ y &= \dfrac{2}{9}\end{align*} $ So, the problem becomes: $ \dfrac{193}{99} \div \dfrac{2}{9} = {?} $ Dividing by a fraction is the same as multiply by the reciprocal of that fraction. $ \dfrac{193}{99} \times \dfrac{9}{2} = {?} $ $ \phantom{\dfrac{193}{99} \times \dfrac{2}{9}} = \dfrac{193 \times 9}{99 \times 2} $ $ \phantom{\dfrac{193}{99} \times \dfrac{2}{9}} = \dfrac{193 \times \cancel{9}} {\cancel{99}11 \times 2} $ $ \phantom{\dfrac{193}{99} \times \dfrac{2}{9}} = \dfrac{193}{22} $